Due to the importance of measuring Pauli observables in near-term applications of quantum computers,
a series of methods [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} have been proposed to reduce the number of measurements needed to estimate
Pauli expectation values.
All of them are based on one basic, yet powerful, observation: Commuting observables can be measured simultaneously.
For example, quantum chemistry applications are often contingent on measuring
\(\mathcal {O}(n^4)\)  Pauli observables [8]}.
The aforementioned Pauli estimation protocols group these observables into \(\mathcal {O}(n^3)\)  or even only \(\mathcal {O}(n)\)  commuting groups.
In turn, \(\mathcal {O}(n^3)\)  or \(\mathcal {O}(n)\)  copies of the underlying state suffice to obtain expectation values for all \(\mathcal {O}(n^4)\)  relevant Pauli observables by exploiting commutation relations.
On the other hand, the novel technique proposed in Appendix 
gets by with even fewer state preparations. A total of \(\mathcal {O}(\log (n^4)) = \mathcal {O}(\log (n))\)  copies suffice.
Restriction to few-body Pauli observables can yield additional improvements. Several protocols are known for this special case, see e.g. [9]}, [10]}, [7]}, [5]}, [13]}, [14]}.
